It is possible to produce an isochronous resonator, for horology, which is independent of position and has a very high quality factor, by combining a pivot having flexible strips with an inertial element, for example a balance.
When such a pivot with flexible strips is made of silicon, the aspect ratio is limited by etching techniques.
The use of a flexible pivot bearing makes it possible replace the conventional contact pivot of a balance, and the elastically returned balance spring. This has the advantage of removing pivot friction, which allows the quality factor of the resonator to be significantly increased. However, flexible pivot bearings are known to:                exhibit parasitic motion of the instantaneous axis of rotation which makes the operation of the resonator sensitive to its position in the gravitational field;        have a non-linear elastic return torque, which makes the resonator anisochronous, i.e. the frequency depends on the amplitude of oscillation;        have a low maximum angular excursion (on the order of 15 to 20 degrees);        have low impact resistance.        
EP Patent Application 2911012 A1 in the name of CSEM describes such a crossed strip flexible pivot, applying the principle stated by W. H. Wittrick, in Aeronautical Quarterly II (1951), wherein the crossing point of the strips is positioned at ⅛th (or ⅞ths) of the length of the strips which notably reduces the effect of positions.
European Patent Application EP14199039.0 in the name of The Swatch Group Research & Development Ltd also describes a resonator with two crossed strips, of the Wittrick pivot type, located in parallel planes and whose crossing point, in projection onto one of these planes, corresponds to the axis of pivoting of the inertial element. This resonator has a specific angle of 72° between the two strips, which optimises the linearity of the elastic return force, and consequently makes the resonator isochronous in a given range of angular amplitude, assuming that it is made of a perfectly elastic material, in other words whose stress depends perfectly linearly on deformation. This particular arrangement makes it possible to optimise the linearity of the elastic return force, and consequently to simultaneously resolve the problem of sensitivity to positions and of anisochronism.
European Patent Application EP16155039.7 in the name of The Swatch Group Research & Development Ltd describes a 2D flexible pivot bearing, i.e. made in one plane, which can easily be made of single crystal silicon. This material has numerous advantages for making a resonator in horological dimensions, in particular excellent elastic properties, precise shaping via photolithography technology, and the possibility of temperature compensation by means of an oxidation layer. Silicon has the drawback, however, of being brittle and therefore sensitive to shocks. This document proposes a different geometry, which makes the resonator simultaneously isochronous and independent of positions, but which has the advantage of being able to be made in a single silicon layer since all the strips are in the same plane.
JP Patent Application 2016133494A in the name of CITIZEN HOLDINGS CO LTD describes a method for manufacturing a double-height spring.
CH Patent Application 709291A2 in the name of CSEM describes an arrangement of crossed flexible strips, of the Wittrick type, with two flexible strips connecting a support element to a balance, capable of exerting a return torque on the balance, wherein a first strip is disposed in a first plane perpendicular to the plane of the oscillator, and a second strip is disposed in a second plane perpendicular to the plane of the oscillator and secant with the first plane, wherein the geometric axis of oscillation of the oscillator is defined by the intersection of the first plane and the second plane, this geometric axis of oscillation crossing the first and second strips at ⅞ths of their respective length.